A calculus of the absurd

Chapter 16 Hyperbolic functions

16.1 Definitions

There are definitely nice ways to think about these (including their relation to hyperbolic geometry, etc.).

The most efficient method is to cut to the chase and define the hyperbolic functions:

\begin{align} \cosh (x) & = \frac {e^{x} + e^{-x}}{2} \\ & = \frac {e^{2x} + 1}{2e^x} \text { multiplying by $\frac {e^{x}}{e^{x}}$} \end{align}

\begin{align} \sinh (x) & = \frac {e^{x} - e^{-x}}{2} \\ & = \frac {e^{2x} - 1}{2e^x} \text { multiplying by $\frac {e^{x}}{e^{x}}$} \end{align}

When plotted, they look like this:

(-tikz- diagram)

(-tikz- diagram)

(-tikz- diagram)